What is Multiple Linear Regression?

Multiple linear regression is the most common form of linear regression analysis.  As a predictive analysis, the multiple linear regression is used to explain the relationship between one continuous dependent variable from two or more independent variables.  The independent variables can be continuous or categorical (dummy coded as appropriate).

Questions Answered:

Do age and IQ scores effectively predict GPA?

How do weight, height, and age explain the variance accounted for in BMI scores?

Assumptions:

Data must be normally distributed

A linear relationship is assumed between the dependent variable and the independent variables.

The residuals are homoscedastic and approximately rectangular-shaped.

Absence of multicollinearity is assumed in the model, so that the independent variables are not too highly correlated.

At the center of the multiple linear regression analysis is the task of fitting a single line through a scatter plot.  More specifically the multiple linear regression fits a line through a multi-dimensional cloud of data points.  The simplest form has one dependent and two independent variables, the general form of the multiple linear regression is defined as

for i = 1…n .

Sometimes the dependent variable is also called endogenous variable, prognostic variable or regressand.  The independent variables are also called exogenous variables, predictor variables or regressors.

There are 3 major uses for Multiple Linear Regression Analysis – (1) causal analysis, (2) forecasting an effect, (3) trend forecasting.  Other than correlation analysis, which focuses on the strength of the relationship between two or more variables, regression analysis assumes a dependence or causal relationship between one or more independent and one dependent variable.

Firstly, it might be used to identify the strength of the effect that the independent variables have on a dependent variable.  Typical questions are what is the strength of relationship between dose and effect, sales and marketing spend, age and income.

Secondly, it can be used to forecast effects or impacts of changes.  That is multiple linear regression analysis helps us to understand how much will the dependent variable change, when we change the independent variables.  Typical questions are how much additional Y do I get for one additional unit X.

Thirdly, multiple linear regression analysis predicts trends and future values.  The multiple linear regression analysis can be used to get point estimates.  Typical questions are what will the price for gold be in 6 month from now?  What is the total effort for a task X?

When selecting the model for the multiple linear regression analysis another important consideration is the model fit.  Adding independent variables to a multiple linear regression model will always increase its statistical validity, because it will always explain a bit more variance (typically expressed as R²).

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Related Pages:

Conduct and Interpret a Multiple Linear Regression

Assumptions of Multiple Linear Regression